Machine Learning and Data Science Seminar

This talk, based on joint work with Alexander Zlokapa, concerns Bayesian inference with neural networks. 

I will begin by presenting a result giving exact non-asymptotic formulas for Bayesian posteriors in deep linear networks. A key takeaway is the appearance of a novel scaling parameter, given by # data * depth / width, which controls the effective depth of the posterior in the limit of large model and dataset size. 

Additionally, I will explain some quite recent results on the role of this effective depth parameter in Bayesian inference with deep non-linear neural networks that have shaped activations.

Data that have an intrinsic network structure can be found in various contexts, including social networks, biological systems (e.g., protein-protein interactions, neuronal networks), information networks (computer networks, wireless sensor networks),  economic networks, etc. As the amount of graphical data that is generated is increasingly large, compressing such data for storage, transmission, or efficient processing has become a topic of interest. In this talk, I will give an information theoretic perspective on graph compression. 

The focus will be on compression limits and their scaling with the size of the graph. For lossless compression, the Shannon entropy gives the fundamental lower limit on the expected length of any compressed representation. I will discuss the entropy of some common random graph models, with a particular emphasis on our results on the random geometric graph model. 

Then, I will talk about the problem of compressing a graph with side information, i.e., when an additional correlated graph is available at the decoder. Turning to lossy compression, where one accepts a certain amount of distortion between the original and reconstructed graphs, I will present theoretical limits to lossy compression that we obtained for the Erdős–Rényi and stochastic block models by using rate-distortion theory.

Low rank structure is pervasive in real-world datasets.

This talk shows how to accelerate the solution of fundamental computational problems, including eigenvalue decomposition, linear system solves, composite convex optimization, and stochastic optimization (including deep learning), by exploiting this low rank structure.

We present a simple method based on randomized numerical linear algebra for efficiently computing approximate top eigende compositions, which can be used to replace large matrices (such as Hessians and constraint matrices) with low rank surrogates that are faster to apply and invert.

The resulting solvers for linear systems (NystromPCG), composite convex optimization (NysADMM), and stochastic optimization (SketchySGD and PROMISE) demonstrate strong theoretical and numerical support, outperforming state-of-the-art methods in terms of speed and robustness to hyperparameters.

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Modern neural network models are trained using fairly standard stochastic gradient optimizers, sometimes employing mild preconditioners. 
A natural question to ask is whether significant improvements in training speed can be obtained through the development of better optimizers. 

In this talk I will argue that this is impossible in the large majority of cases, which explains why this area of research has stagnated. I will go on to identify several situations where improved preconditioners can still deliver significant speedups, including exotic architectures and loss functions, and large batch training. 

TBA